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%I #7 Mar 26 2018 20:02:12
%S 1,1,1,4,1,4,1,10,4,4,1,16,1,4,4,34,1,16,1,16,4,4,1,54,4,4,10,16,1,22,
%T 1,80,4,4,4,78,1,4,4,54,1,22,1,16,16,4,1,181,4,16,4,16,1,54,4,54,4,4,
%U 1,102,1,4,16,254,4,22,1,16,4,22,1,272,1,4,16,16
%N Number of thrice-factorizations of n.
%C A thrice-factorization of n is a choice of a twice-factorization of each factor in a factorization of n. Thrice-factorizations correspond to intervals in the lattice form of the multiorder of integer factorizations.
%H Gus Wiseman, <a href="/A301598/a301598.png">The (2*2*3*3) component of the lattice form of the multiorder of integer factorizations.</a>
%H Gus Wiseman, <a href="/A301598/a301598_1.png">The (2*2*3*3) component, version 2.</a>
%F Dirichlet g.f.: Product_{n > 1} 1/(1 - A281113(n)/n^s).
%e The a(12) = 16 thrice-factorizations:
%e ((2))*((2))*((3)), ((2))*((2)*(3)), ((3))*((2)*(2)), ((2)*(2)*(3)),
%e ((2))*((2*3)), ((2)*(2*3)),
%e ((2))*((6)), ((2)*(6)),
%e ((3))*((2*2)), ((3)*(2*2)),
%e ((3))*((4)), ((3)*(4)),
%e ((2*2*3)),
%e ((2*6)),
%e ((3*4)),
%e ((12)).
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t twifacs[n_]:=Join@@Table[Tuples[facs/@f],{f,facs[n]}];
%t thrifacs[n_]:=Join@@Table[Tuples[twifacs/@f],{f,facs[n]}];
%t Table[Length[thrifacs[n]],{n,15}]
%Y Cf. A001055, A007716, A050336, A050338, A063834, A162247, A269134, A281113, A281116, A301595, A301598, A301706.
%K nonn
%O 1,4
%A _Gus Wiseman_, Mar 24 2018
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